Here \(X^{(r)l}, Y^{(r)m}_{\mu }\)  and \(Z^{(r)n}_{\mu \nu }\)  are the \(r^{th}\)  order scalar, vector and rank-2 tensor thermodynamic forces, respectively. The indices \(l, m\) , and \(n\)  denote different
independent thermodynamic forces of each tensorial rank for each order \(r\) . \(A_{l}^{(r)}, B_{m}^{(r)\mu }\)  and \(C_{n}^{(r)\mu \nu }\)  are the unknown coefficients functions of space-time, particle momentum
and the ratio of it's rest mass to the temperature \(z=m/T\) . It is customary to expand the coefficients in a polynomial basis to extract their
values as [1]}
\(A_{l}^{(r)}=\sum _{s=0}^{\infty }A_{l}^{r,s}(z, x) \tilde{E}_{\text{p}}^{s}, B_{m}^{(r) \mu }=\sum _{s=0}^{\infty } B_{m}^{r,s}(z, x) \tilde{E}_{\text{p}}^{s} \tilde{p}^{\langle \mu \rangle }\) ,
\(C_{n}^{(r) \mu \nu }=\sum _{s=0}^{\infty } C_{n}^{r,s}(z, x)\tilde{E}_{\text{p}}^{s} \tilde{p}^{\langle \mu } \tilde{p}^{\nu \rangle }\) .
Here \(\tilde{p}^{\mu }=p^{\mu }/T\) , \(\tilde{\mu }=\mu /T\) , and \(\tilde{E}_{\text{p}} = E_{\text{p}}/T\) ; where \(E_{\text{p}} = u_{\mu }p^{\mu }\) , \(u^{\mu }\)  is the fluid four velocity
of the system at equilibrium. We used, \(A_{\langle \mu \rangle }=\Delta _{\mu \nu }A^{\nu }\) , \( A_{\langle \mu } B_{\nu \rangle } =\Delta _{\mu \nu }^{\alpha \beta }A_{\alpha }B_{\beta }\)  and \(\Delta ^{\mu \nu }=g^{\mu \nu }-u^{\mu }u^{\nu }\) .
